The introduction of an excellent paper by Dan Waxman called 'Deflationism, Arithmetic, and the Argument from Conservativeness', forthcoming in Mind (draft here), made me think about this.

In his initial, rough gloss of deflationism, after mentioning the correspondence and the coherence theories, Waxman writes that '[u]nlike these theories and others like them, deflationism rejects the idea that truth plays any serious explanatory role in philosophy or elsewhere'.

One of the things you can do with formulae of formal logic is use them to make statements about how things are. This may not be their main point, but it is undoubtedly something you can do. What ways are there to do it? I will take the propositional calculus as a case study and consider three existing ways, before introducing a new one.

Early on in modern logic, one common way to use the formulae of the propositional calculus to make statements was to have the atomic formulae (and sometimes connectives too) denote or refer to things, such as sentences or propositions. This sometimes led to use-mention confusions, but there are ways around that.

This post is, in a way, against logical pluralism. But the form of monism about the consequence relation I want to put forward is unlike existing, or at least the most visible, contenders. It is inspired by Colin R. Caret and Ole Hjortland's first chapter to the new volume Foundations of Logical Consequence. (The chapter has just appeared on Academia.edu.)